Chapter 14. Fatigue of Materials:

Seoul National University

Seoul National University System Health & Risk Management

- 1 -

Chapter 14. Fatigue of Materials:

Strain-Based Approach to Fatigue

Seoul National University System Health & Risk Management

Jong Moon Ha

2015. 05. 06

CHAPTER 14 Objectives

• Explorer strain versus fatigue life curves and equations, including trends with material and adjustments for surface finish and size

• Extend strain-life curves to cases of nonzero mean stress and multi-axial stress

• Apply the strain-based method to make life estimates for engineering

components, especially members with geometric notches, including

cases of irregular variation of load with time

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* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

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Review: Crack Initiation and Propagation*

• The fatigue life of a component is made up of initiation and propagation stages

• The size of crack is usually unknown and often depends on the size of the component being analyzed (e.g. A: 0.1mm-crack, B:0.15mm-crack)

• Stress & Strain-based Approach: To determine crack initiation life

• Fracture Mechanics: To estimate the crack propagation life

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273.

Figure 3.1* Initiation and propagation

portions of fatigue life Figure 3.9* Extended service life of a cracked component

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14.1 Introduction – Why strain-based approach?

Brief Review of Stress-based Approach

• Key Idea is to develop stress-life relationship by employing elastic stress concentration factors and empirical modifications thereof.

• Stress-based approach emphasizes nominal stress, rather than local stresses and strains

• Stress-based approach does not account for plastic strain.

Figure 9.5

Stress-based Approach vs Strain-based Approach

High cycle fatigue Low cycle fatigue

Elastic range of the material

(Does not account for plastic strain) Account for plastic strain More than 1000 cycles Less than 1000 cycles Stress-Life (Stress controlled) Strain-Life (Strain controlled)

Source: http://www.public.iastate.edu/~gkstarns/

14.1 Introduction

• Employment of cyclic stress-strain curve is a unique feature of the strain-based approach, as is the use of a strain versus life curve, instead of a nominal stress versus life (S-N) curve

• Strain-based method gives improved estimates for intermediate and especially short fatigue lives.

• Strain-based approach employs the local mean stress at the notch, rather than the mean nominal stress.

• Certain concepts employed will be related to those introduced in Chapter 9 and 10.

Also, we will draw upon the information in Chapter 12 and 13 on plastic deformation

 Be familiar with the contents in Chapter 9, 10, 12 and 13 before studying Chapter 14.

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14.2 Strain Versus Life Curves

• A strain versus life curve is a plot of strain amplitude versus cycles to failure (See Fig. 14.3)

Figure 14.3 Elastic, plastic, and total strain versus life curves. (Adapted from [Landgraf 70]; copyright © ASTM;

reprinted with permission.)

14.2.1 Strain-Life Tests and Equations

• Of particular relevance is the cyclic stress-strain curve (Also can be found in Eq. 12.54)

𝜀𝜀_𝑎𝑎 = σ_𝑎𝑎 𝐸𝐸 +

σ_𝑎𝑎 𝐻𝐻^′

1/𝑛𝑛^′

(14.1)

𝜀𝜀_𝑎𝑎 = 𝜀𝜀_{𝑒𝑒𝑎𝑎}+𝜀𝜀_{𝑝𝑝𝑎𝑎} (14.2)

Measure of the half-width of the stress-strain hysteresis loop

• Note that the strain amplitude can be divided into elastic and plastic parts as

𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_𝑎𝑎 𝐸𝐸

• Strain-life curves are derived from fatigue tests under completely reversed cyclic loading between constant strain limits (See Fig. 14.2)

Figure 14.2

Measurement:

𝜎𝜎_𝑎𝑎, 𝜀𝜀_𝑎𝑎, 𝜀𝜀_{𝑝𝑝𝑎𝑎}

Figure 12.17

𝜎𝜎_𝑎𝑎 = ∆𝜎𝜎/2 𝜀𝜀_𝑎𝑎 = ∆𝜀𝜀/2

14.2.1 Strain-Life Tests and Equations

• For each test, three points are plotted (Fig. 14.4)

• If data from several tests are plotted, the elastic strains often give a straight line of shallow slope on a log-log plot, and the plastic strains give a straight line of steeper slope.

• Equation can then be fitted to these lines 𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_𝑎𝑎

𝐸𝐸 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏

𝜀𝜀_{𝑝𝑝𝑎𝑎} = 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐

Figure 14.4 Strain versus life curves for RQC- 100 steel. For each of several tests, elastic, plastic, and total strain data points are plotted versus life, and fitted lines are also shown.

(From the author’s data on the ASTM Committee E9 material.)

(14.3) – (a) (14.3) – (b)

• Four constants(σ_𝑓𝑓^′, 𝑏𝑏, 𝜀𝜀_𝑓𝑓^′, 𝑐𝑐) are considered to be material properties

• Coffin-Manson Relationship 𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.4)

①

③

②

Plastic Elastic

(𝜎𝜎_𝑎𝑎/𝐸𝐸)

Remind - 14.2 Strain Versus Life Curves

Figure 14.3 Elastic, plastic, and total strain versus life curves. (Adapted from [Landgraf 70]; copyright © ASTM;

reprinted with permission.)

𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.4)

Four constants (σ_𝑓𝑓^′, 𝑏𝑏, 𝜀𝜀_𝑓𝑓^′, 𝑐𝑐)

• Coffin-Manson Relationship

Plastic Elastic

14.2.3 Trends for Engineering Metals

• At long lives: the curve approaches the elastic strain line

• At short lives: the curve approaches the plastic strain line

• Near the crossing point: the two types of strain are of similar magnitude

 Transition Fatigue Life, 𝑁𝑁_𝑡𝑡

• 𝑁𝑁_𝑡𝑡 is the most logical point for separating low-cycle fatigue and high-cycle fatigue

• Special analysis of plasticity effects by the strain-based approach may be needed if lives around or less than 𝑵𝑵_𝒕𝒕 are of interest

𝑁𝑁_𝑡𝑡 = 1 2

𝜎𝜎_𝑓𝑓^′ 𝜀𝜀_𝑓𝑓^′𝐸𝐸

1/(𝑐𝑐−𝑏𝑏)

Let 𝜀𝜀_{𝑒𝑒𝑎𝑎} = 𝜀𝜀_{𝑝𝑝𝑎𝑎}

(14.6)

Strain-based Approach

14.2.3 Trends for Engineering Metals

• The strain-life equation requires four empirical constants (σ_𝑓𝑓^′, 𝑏𝑏, 𝜀𝜀_𝑓𝑓^′, 𝑐𝑐)

• Several points must be considered in attempting to obtain these constants from fatigue data*

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

1) Generalization

Not all materials may be represented by the four-parameter strain-life equation.

(Examples: some high strength aluminum alloys and titanium alloys) 2) Data Size

The four fatigue constants may represent a curve fit to a limited number of data points. They may be changed if more data points are included in the curve fit.

3) Range of data

The fatigue constants are determined from a set of data points over a given range.

Gross error may occur when extrapolating fatigue life estimates outside this range.

4) Physical phenomenon

The use of this equation is strictly a matter of mathematical convenience and is not based on a physical phenomenon

Nevertheless, the following approximate methods may be useful (cont.).

14.2.3 Trends for Engineering Metals

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p3 64.

𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐

Fatigue Strength Coefficient, 𝝈𝝈_𝒇𝒇^′ Fatigue Ductility Coefficient, 𝜺𝜺_𝒇𝒇^′ Approximation* σ^𝑓𝑓′ ≈ 𝜎𝜎_𝑓𝑓 (corrected for necking)

For steels (<500 BHN): σ_𝑓𝑓^′ ≈ 𝑆𝑆_𝑢𝑢 + 50𝑘𝑘𝑘𝑘𝑘𝑘 𝜀𝜀_𝑓𝑓^′ ≈ 𝜀𝜀_𝑓𝑓, where 𝜀𝜀_𝑓𝑓 = ln _1−RA¹

RA is the reduction in area

A ductile Metal Low High

A brittle metal High Low

• All pass near the strain 𝜀𝜀_𝑎𝑎 = 0.01 for a life of 𝑁𝑁_𝑓𝑓 = 1000 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑘𝑘

(Brittle) High plasticity

Figure 14.6 Trends in strain–life curves for strong, tough, and ductile metals. (Adapted from [Landgraf 70]; copyright © ASTM; reprinted with permission.)

14.2.3 Trends for Engineering Metals

• Fatigue Strength Exponent, b Average: -0.085

For soft metals: 𝑏𝑏 ≈ −0.12

For highly hardened metals: 𝑏𝑏 ≈ −0.05

For steels with ultimate tensile strengths below about 𝜎𝜎_𝑢𝑢 = 1400MPa

(Fatigue limit occurs near 𝑁𝑁_𝑓𝑓 = 10⁶ cycles at a stress amplitude around 𝜎𝜎_𝑎𝑎 = 𝜎𝜎_𝑢𝑢/2)

In other case, where the fatigue limit (or long-life fatigue strength) at cycles is given by 𝜎𝜎_𝑎𝑎 = 𝑚𝑚_𝑒𝑒𝜎𝜎_𝑢𝑢, the estimate becomes

𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_𝑎𝑎 𝐸𝐸 =

σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 (14.3) – (a)

𝑏𝑏 = − 1

𝑐𝑐𝑙𝑙𝑙𝑙 2𝑁𝑁_𝑓𝑓 𝑐𝑐𝑙𝑙𝑙𝑙 2σ_𝑓𝑓^′

σ_𝑢𝑢 = − 1

6.3 𝑐𝑐𝑙𝑙𝑙𝑙

2�σ_𝑓𝑓

σ_𝑢𝑢 (14.10)

𝑏𝑏 = − 1

𝑐𝑐𝑙𝑙𝑙𝑙 2𝑁𝑁_𝑓𝑓 𝑐𝑐𝑙𝑙𝑙𝑙 �σ_𝑓𝑓

𝑚𝑚_𝑒𝑒σ_𝑢𝑢 (14.11)

𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐

14.2.3 Trends for Engineering Metals

• Fatigue Ductility Exponent, 𝑐𝑐 *

𝑐𝑐 is not well defined as the other parameters.

A rule of thumb approach must be followed rather than an empirical equation Coffin found 𝑐𝑐 to be about -0.5

Manson found 𝑐𝑐 to be about -0.6

Morrow found that 𝑐𝑐 varied between -0.5 and -0.7

Fairly ductile metals (where 𝜀𝜀_𝑓𝑓 ≈ 1) have average values of 𝑐𝑐=-0.6 For strong metals (where 𝜀𝜀_𝑓𝑓 ≈ 0.5) have average values of 𝑐𝑐=-0.5

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p3 64.

𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐

14.2.3 Trends for Engineering Metals

• Hardness varies inversely with ductility

so that N_t decreases as hardness is increased.

Figure 14.8 Transition fatigue life versus hardness for a wide range of steels. (Adapted from [Landgraf 70];

copyright © ASTM; reprinted with permission.)

14.2.4 Factors Affecting Strain-Life Curves

• Is a hostile chemical environment or elevated temperature is present, smaller numbers of cycles to failure are expected

• Temperature

: At temperature exceeding about half of the absolute melting temperature of a given material, nonlinear deformation due to time-dependent creep-relaxation behavior generally become significant

• Residual Stress

Residual stress are quickly removed by cycle-dependent relaxation if cyclic plastic strains are present

 These have only limited effect at lives around and below N_t. Some additional comments on this topic are given in Chapter 15.

14.2.4 Factors Affecting Strain-Life Curves : Surface Finish

• High-cycle fatigue (most of the life is spent initiating a crack)

 Surface finish is important

• Fatigue with significant plastic strain (most of the life is spent growing crack size)

 Surface finish cannot have an effect

• A reasonable method to include the effect of surface finish is to change only the elastic slope b.

• When the fatigue limit at 𝑁𝑁_𝑐𝑐 cycles is given by 𝜎𝜎_𝑎𝑎 = 𝑚𝑚_𝑠𝑠𝜎𝜎_𝑒𝑒 , where 𝑚𝑚_𝑠𝑠 is a surface effect factor, as in Chapter 10

𝑏𝑏_𝑠𝑠 = − 1

𝑐𝑐𝑙𝑙𝑙𝑙 2𝑁𝑁_𝑓𝑓 𝑐𝑐𝑙𝑙𝑙𝑙 �σ_𝑓𝑓

𝑚𝑚_𝑠𝑠σ_𝑒𝑒 = 𝑏𝑏 + 𝑐𝑐𝑙𝑙𝑙𝑙 𝑚𝑚_𝑠𝑠

𝑐𝑐𝑙𝑙𝑙𝑙 2𝑁𝑁_𝑒𝑒 (14.12) 𝜎𝜎_𝑒𝑒 = 𝜎𝜎_𝑓𝑓^′ 2𝑁𝑁_𝑒𝑒 ^𝑏𝑏

14.2.4 Factors Affecting Strain-Life Curves : Size Effect

• Size Effect (as discussed in Chapter 10) are also a concern in applying a strain- based approach to large-size members, but experimental data are limited

• A study suggested lowering the entire strain-life curve by this factors so that the intercept constants 𝜎𝜎_𝑓𝑓^′ and 𝜀𝜀_𝑓𝑓^′ are replaced by reduced values and 𝜎𝜎_{𝑓𝑓𝑑𝑑}^′ and 𝜀𝜀_{𝑓𝑓𝑑𝑑}^′

where 𝑚𝑚_𝑑𝑑 for the shafts up to 200 mm in diameter (low-carbon and low-alloy steels) was found to vary with shaft diameter as

(14.14) 𝜎𝜎_{𝑓𝑓𝑑𝑑}^′ = 𝑚𝑚_𝑑𝑑𝜎𝜎_𝑓𝑓^′, 𝜀𝜀_{𝑓𝑓𝑑𝑑}^′ = 𝑚𝑚_𝑑𝑑𝜀𝜀_𝑓𝑓^′

(14.13) 𝑚𝑚_𝑑𝑑 = 𝑑𝑑

25.4𝑚𝑚𝑚𝑚

−0.093

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14.3.1 Mean Stress Effects*

• Cyclic fatigue properties of a material are obtained from completely reversed, constant amplitude strain-controlled tests.

However, components seldom experience this type of loading.

• Mean stress effect can either increase the fatigue life with a nominally compressive load or decrease it with a nominally tensile value (Fig. 2.21*)

• At high strain amplitude (0.5% to 1% or above), where plastic strains are

significant, mean stress relaxation occurs and the mean stress tends toward zero

• Modifications to the strain-life equation have been made to account for mean stress effects by 1) Morrow, 2) Smith, Watson, and Topper (SWT), and 3) Walker.

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

Figure 2.21* Effect of mean stress on strain-life curve

14.3.3 Mean Stress Equation of Morrow*

• Morrow (1968) suggested to modify the elastic term in the strain-life equation

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_𝑎𝑎 𝐸𝐸 =

σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏

(14.3) – (a) 𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_𝑎𝑎 𝐸𝐸 =

σ_𝑓𝑓^′ − σ_𝑚𝑚

𝐸𝐸 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐

(14.4)

𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′ − σ_𝑚𝑚

𝐸𝐸 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐

(14.27)

Figure 2.23* Morrow’s mean stress correction to

the strain-life curve for a tensile mean Figure 14.12 Family of strain–life curves given by the modified Morrow approach

14.3.3 Mean Stress Equation of Morrow (modified)*

• Morrow and Halford (1981) found that ratio of elastic to plastic strain in the equation proposed by Morrow (1968) is dependent on mean stress, which is not true.

• They modified both the elastic and plastic terms of the strain-life equation to maintain the independence of the elastic-plastic strain ratio from mean stress as:

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′ − σ_𝑚𝑚

𝐸𝐸 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ σ_𝑓𝑓^′ − σ_𝑚𝑚 σ_𝑓𝑓^′

𝑐𝑐/𝑏𝑏

2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.24)

• Refer to the Appendix for more details

Figure 2.25* Mean stress correction for independence of elastic/plastic strain ratio from mean stress

(SWT) Parameters, Walker Mean Stress Equation

14.3.5 Smith, Watson, and Topper (SWT) Parameters 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^{′ 2}

𝐸𝐸 2𝑁𝑁_𝑓𝑓 ^2𝑏𝑏 + σ_𝑓𝑓^′𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^{𝑏𝑏+𝑐𝑐} (14.29)

where 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} = 𝜎𝜎_𝑚𝑚 + 𝜎𝜎_𝑎𝑎

14.3.5 Walker Mean Stress Equation 𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸

1 − 𝑅𝑅 2

(1−𝛾𝛾)/𝑏𝑏

2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 1 − 𝑅𝑅 2

𝑐𝑐(1−𝛾𝛾)/𝑏𝑏

2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.33)

14.3.5 Smith, Watson, and Topper (SWT) Parameters

Figure 14.13 Plot of the Smith, Watson, and Topper parameter versus life for the data of Fig. 14.10.

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• Fatigue under multi-axial loading where plastic deformations occur is currently an area of active research.

• Reasonable estimates are possible for relatively simple situations

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14.4.1 Effective Strain Approach

• Consider situations where all cyclic loadings have the same frequency and are either in-phase or 180° out-of-phase.

• Recall three-dimensional stress-strain relationships (Ch. 12)

̅𝜀𝜀 = �𝜎𝜎

𝐸𝐸 + ̅𝜀𝜀^𝑝𝑝 (12.23)

̅𝜀𝜀𝑝𝑝 = 2/3 × 𝜀𝜀_𝑝𝑝1 − 𝜀𝜀_𝑝𝑝2 ² + 𝜀𝜀_𝑝𝑝2 − 𝜀𝜀_𝑝𝑝3 ² + 𝜀𝜀_𝑝𝑝3 − 𝜀𝜀_𝑝𝑝1 ² (12.22)

• Then, we can define an effective strain amplitude

�𝜎𝜎 = 1/ 2 × 𝜎𝜎₁ − 𝜎𝜎₂ ² + 𝜎𝜎₂ − 𝜎𝜎₃ ² + 𝜎𝜎₃ − 𝜎𝜎₁ ² (12.21)

̅𝜀𝜀𝑎𝑎 = �𝜎𝜎_𝑎𝑎

𝐸𝐸 + ̅𝜀𝜀^{𝑝𝑝𝑎𝑎} (14.34)

14.4.1 Effective Strain Approach

̅𝜀𝜀𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.35)

�𝜎𝜎_𝑎𝑎 = σ_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏, _{𝑝𝑝𝑎𝑎}̅𝜀𝜀 = 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 (14.36)

• Consider the special case of plane stress (Refer to Ch. 12.3.4 Application to Plane Stress) 𝜎𝜎_2𝑎𝑎 = 𝜆𝜆𝜎𝜎_1𝑎𝑎, 𝜎𝜎_3𝑎𝑎 = 0, 𝜀𝜀_1𝑎𝑎 = 𝜀𝜀_{𝑒𝑒1𝑎𝑎} + 𝜀𝜀_{𝑝𝑝1𝑎𝑎}

• Combining Eqs. (12.19), (12.24) and (12.32) with Eqs. (14.35), (14.37)

(14.37)

𝜀𝜀_1𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 1 − 𝜈𝜈𝜆𝜆 2𝑁𝑁^{𝑓𝑓 𝑏𝑏} + 𝜀𝜀_𝑓𝑓^′ 1 − 0.5𝜆𝜆 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐

1 − 𝜆𝜆 + 𝜆𝜆² (14.38)

• For the special state of plane stress that is pure shear (𝜎𝜎_2𝑎𝑎 = 𝜆𝜆𝜎𝜎_1𝑎𝑎, 𝜆𝜆 = −1)

• For uniaxial loading (𝜎𝜎_𝟐𝟐 = 𝜎𝜎_𝟑𝟑 = 0)

its value reduces to the uniaxial strain amplitude ( ̅𝜀𝜀_𝑎𝑎 = 𝜀𝜀_1𝒂𝒂)

𝛾𝛾_{𝑚𝑚𝑥𝑥𝑎𝑎} = σ_𝑓𝑓^′

3𝐺𝐺 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 + 3𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.39)

Shear strain amplitude Shear modulus Refer to Ch. 13.4 for more information

14.4.2 Discussion of the Effective Strain Approach

• Consider the hydrostatic stress

(14.40) 𝜎𝜎_ℎ𝑎𝑎 = 𝜎𝜎_1𝑎𝑎 + 𝜎𝜎_2𝑎𝑎 + 𝜎𝜎_3𝑎𝑎

3

• Relative value of 𝜎𝜎_ℎ𝑎𝑎 my be expressed as a triaxiality factor for plane stress (𝜎𝜎_3𝑎𝑎 = 0) 𝑇𝑇 = 1 + 𝜆𝜆

1 − 𝜆𝜆 + 𝜆𝜆² , (14.41)

(a) Pure planar shear (𝜆𝜆 = −1)  𝑇𝑇 = 0 (b) Uniaxial stress (𝜆𝜆 = 0)  𝑇𝑇 = 1 (c) Equal biaxial stress (𝜆𝜆 = 1)  𝑇𝑇 = 2 𝜎𝜎_2𝑎𝑎 = 𝜆𝜆𝜎𝜎_1𝑎𝑎,

• Marloff (1985) proposed to include this effect to strain-life equation as:

̅𝜀𝜀𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 2^1−𝑇𝑇𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.42)

14.4.3 Critical Plane Approaches

• Critical plane approach is needed where the loading is non-proportional to a significant degree

• Stresses and strains normal to the crack plane may have a major effect on the behavior, accelerating the growth if they tend to open the crack.

1) Stresses and strains are determined for various orientations (planes) in the material.

2) Stresses and strains acting on the most severely loaded plane are used for analysis.

14.4.3 Critical Plane Approaches

• Fatemi and Socie 𝛾𝛾_{𝑎𝑎𝑐𝑐} 1 + 𝛼𝛼𝜎𝜎_max𝑐𝑐

σ_𝑜𝑜^′ _𝑎𝑎̅𝜀𝜀 = 𝜏𝜏_𝑓𝑓^′

𝐺𝐺 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝛾𝛾_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.43)

where 𝛾𝛾_{𝑎𝑎𝑐𝑐} is the largest amplitude of shear strain for any plane, 𝜎𝜎_max𝑐𝑐 is the peak tensile stress normal to the plane of 𝛾𝛾_{𝑎𝑎𝑐𝑐}, 𝛼𝛼 is an empirical constant ranging from 0.6 to 1.0, and σ_𝑜𝑜^′ is the yield strength for the cyclic stress-strain curve.

𝜏𝜏_𝑓𝑓^′, 𝑏𝑏, 𝛾𝛾_𝑓𝑓^′, 𝑐𝑐 give the strain-life curve from completely reversed tests in pure shear.

(specifically torsion tests on thin-walled tubes)

Figure 14.14 Crack under pure shear (a), where irregularities retard growth, compared with a situation (b) where a normal stress acts to open the crack, enhancing its growth. (Adapted from [Socie 87]; used with permission of ASME.)

14.4.3 Critical Plane Approaches

• Fatemi and Socie

(14.35)

• Smith, Watson, and Topper (SWT) Parameters 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^{′ 2}

𝐸𝐸 2𝑁𝑁_𝑓𝑓 ^2𝑏𝑏 + σ_𝑓𝑓^′𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^{𝑏𝑏+𝑐𝑐}

 The shortest life estimated from either Eqs. (14.43) or (14.35) is the final life estimate.

𝛾𝛾_{𝑎𝑎𝑐𝑐} 1 + 𝛼𝛼𝜎𝜎_max𝑐𝑐

σ_𝑜𝑜^′ _𝑎𝑎̅𝜀𝜀 = 𝜏𝜏_𝑓𝑓^′

𝐺𝐺 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝛾𝛾_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.43)

• A single multiaxial fatigue criterion that considers both the shear and normal stress cracking mode is that of Chu (1995)

(14.44) 2𝜏𝜏_{𝑚𝑚𝑎𝑎𝑚𝑚}𝛾𝛾_𝑎𝑎 + 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_𝑎𝑎 = 𝑓𝑓 𝑁𝑁_𝑓𝑓

where 𝑓𝑓 𝑁𝑁_𝑓𝑓 can be obtained from uniaxial test data.

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14.5.1 Constant Amplitude Loading

• Assuming idealized behavior for the material

𝜀𝜀 = 𝑓𝑓 𝜎𝜎 , 𝜀𝜀_𝑎𝑎 = 𝑓𝑓(𝜎𝜎_𝑎𝑎) (14.45)

(Monotonic) (Cyclic)

∆𝜀𝜀

2 = 𝑓𝑓

∆𝜎𝜎 2

(14.46)

Figure 12.14 Stress–strain unloading and reloading behavior consistent with a spring and slider rheological model.

The example curves plotted correspond to a Ramberg–Osgood stress–strain curve with constants as in Fig. 12.9.

14.5.1 Constant Amplitude Loading

• Consider Neuber’s rule to analyze a notched member

 Ch. 13 will be reviewed in the following two slides

Ch. 13 Stress-strain Analysis of Plastically deforming Members

13.5.3 Estimates of Notch Stress and Strain for Local Yielding

• Theoretical stress concentration factor, 𝐾𝐾_𝑡𝑡, is used to relate the nominal stress or strain to the local values.*

• Upon yielding, the local values are no longer linearly related to the nominal values by 𝐾𝐾_𝑡𝑡*

• Instead, the values are related in terms of stress and strain concentration factors as:

𝑘𝑘_𝜎𝜎 = 𝜎𝜎 (Local stress)

S (Nominal stress) , 𝑘𝑘^𝜀𝜀 = 𝜀𝜀 (Local strain)

e (Nominal strain) (13.56)

Figure 14.13* Effect of yielding on 𝐾𝐾_𝜎𝜎 and 𝐾𝐾_𝜖𝜖

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

• After yielding, the actual local stress is less than that predicted using 𝐾𝐾_𝑡𝑡

• After yielding, the actual local strain is greater than that predicted using 𝐾𝐾_𝑡𝑡

• This is due to residual stresses at the notch root

Ch. 13 Stress-strain Analysis of Plastically deforming Members

13.5.3 Estimates of Notch Stress and Strain for Local Yielding

• Neuber’s rule states simply that the geometric mean of the stress and strain concentration factors remains equal to 𝑘𝑘_𝑡𝑡 during plastic deformation

• If fully plastic yielding does not occur, e=S/E ^applies.

𝑘𝑘_𝑡𝑡 = 𝑘𝑘_𝜎𝜎𝑘𝑘_𝜀𝜀 (13.57)

𝜎𝜎𝜀𝜀 = 𝑘𝑘_𝑡𝑡𝑆𝑆 ²

E ^(13.58)

Figure 13.16 For a given notched member and stress–strain curve (a), Neuber’s rule may be used to estimate local notch stresses and strains, σ and ε, corresponding to a particular value of nominal stress S. Stress and strain concentration factors vary as in (b).

14.5.1 Constant Amplitude Loading

• Consider Neuber’s rule to analyze a notched member Step 1

Step 2

Step 3 Step 4

Figure 14.15 Steps required in strain-based life prediction for a notched member under constant amplitude loading.

14.5.1 Constant Amplitude Loading

• Step 1

14.5.1 Constant Amplitude Loading

• Step 2 & Step 3

𝜀𝜀 = σ 𝐸𝐸 +

σ 𝐻𝐻^′

1/𝑛𝑛^′

𝜎𝜎𝜀𝜀 = 𝑘𝑘_𝑡𝑡𝑆𝑆 ² E

14.5.1 Constant Amplitude Loading

• Step 4

14.5.2 Irregular Load Versus Time Histories

• The life to failure N_f corresponding to each hysteresis loop can be determined from its combination of strain amplitude and mean stress.

• If the SWT parameter is used, 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} is simply the highest stress (𝜎𝜎_𝐺𝐺) for F-G-F’

• Having obtained the Nf value for each loop, we can apply the Palmgren Miner rule (Ch.

9), where each closed stress-strain hysteresis loop is considered to represent a cycle.

Figure 14.16 Analysis of a notched member subjected to an irregular load versus time history. Notched member (a), made of 2024-T351 aluminum, is subjected to load history (b). The resulting local stress–strain response at the notch is shown in (c).

F-G-F’

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6. Comparison of Methods*

General Points for Comparison

• Are the methods to be used in the design cycle or to analyze an existing component?

• What is the accuracy of the methods compared to input variables such as load history and material properties?

• What are the relative economics?

• What is the level of acceptance?

• Uses in design versus research.

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

6. Comparison of Methods*

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

Stress-based Approach Strain-based Approach

Economics Quick and Cheap Expensive

Available Data Size Big Small

History Long (100 years) Short (30 years)

Amount of Confidence High Low

Complexity Low High

Physical Insight Bad Good

Plastic Strain Not considered Considered

Residual Stress Not considered Considered

Where should we

use this approach? • For constant amplitude loading and long fatigue lives.

• When elastic strains are dominant (e.g. Transmission shaft, valve springs, and gears)

• For variable amplitude loading and short fatigue lives.

• When plastic strains are significant

• High temperature applications with fatigue-creep (e.g. Gas turbine engine)

Appendix

14.3.2 Including Mean Stress Effects in Strain-Life Equation

• Let us define equivalent completely reversed stress amplitude 𝜎𝜎_{𝑎𝑎𝑎𝑎} (as Eq. 9.22), which is repeated here:

(14.15) 𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝜎𝜎_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏

• An additional equation is needed to calculate 𝜎𝜎_{𝑎𝑎𝑎𝑎} for the mean stress situation (14.17)

𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝑓𝑓 𝜎𝜎_𝑎𝑎, 𝜎𝜎_𝑚𝑚 = 𝜎𝜎_𝑎𝑎 𝑓𝑓 𝜎𝜎_𝑎𝑎, 𝜎𝜎_𝑚𝑚

𝜎𝜎_𝑎𝑎 = 𝜎𝜎_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏

• Then, we can define zero-mean-stress-equivalent life, N*

𝜎𝜎_𝑎𝑎 = 𝜎𝜎_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 𝜎𝜎_𝑎𝑎 𝑓𝑓 𝜎𝜎_𝑎𝑎, 𝜎𝜎_𝑚𝑚

1/𝑏𝑏 𝑏𝑏

= 𝜎𝜎_𝑓𝑓^′ 2𝑁𝑁^{∗ 𝑏𝑏}

where 𝑁𝑁^∗ = 𝑁𝑁_{𝑓𝑓 𝑓𝑓 𝜎𝜎}^{𝜎𝜎𝑎𝑎}

𝑎𝑎,𝜎𝜎_𝑚𝑚

1/𝑏𝑏

(14.18)

14.3.3 Mean Stress Equation of Morrow*

• Equivalent completely reversed stress amplitude (Also can be found at Eq. (9.21)) (14.21)

𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝜎𝜎_𝑎𝑎 1 − 𝜎𝜎𝜎𝜎^𝑚𝑚_𝑓𝑓^′

𝑁𝑁_{𝑚𝑚𝑚𝑚}^∗ = 𝑁𝑁_𝑓𝑓 1 − 𝜎𝜎_𝑚𝑚 𝜎𝜎_𝑓𝑓^′

1/𝑏𝑏 (14.22)

𝑁𝑁_𝑓𝑓 = 𝑁𝑁_{𝑚𝑚𝑚𝑚}^∗ 1 − 𝜎𝜎_𝑚𝑚 𝜎𝜎_𝑓𝑓^′

−1/𝑏𝑏

(14.23)

𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^{∗ 𝑏𝑏} + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁^{∗ 𝑐𝑐} 𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′

𝐸𝐸 1 − 𝜎𝜎_𝑚𝑚

𝜎𝜎_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 1 − 𝜎𝜎_𝑚𝑚 𝜎𝜎_𝑓𝑓^′

𝑐𝑐/𝑏𝑏

2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.24)

14.3.3 Mean Stress Equation of Morrow*

Figure 14.11 Mean stress data of Fig. 14.10 plotted versus N^∗ according to the Morrow equation.

14.3.5 Smith, Watson, and Topper (SWT) Parameters

• This approach assumes that the life for any situation of mean stress depends on the product

where 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} = 𝜎𝜎_𝑚𝑚 + 𝜎𝜎_𝑎𝑎 and ℎ^′′ 𝑁𝑁_𝑓𝑓 indicates a function of fatigue life 𝑁𝑁_𝑓𝑓

• Life is expected to be the same as for completely reversed loading where this product has the same value.

• Let σ_{𝑎𝑎𝑎𝑎} and 𝜀𝜀_{𝑎𝑎𝑎𝑎} be the completely reversed stress and strain amplitude that result in the same life 𝑁𝑁_𝑓𝑓 as the 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}, 𝜀𝜀_𝑎𝑎 combination.

• When 𝜎𝜎_𝑚𝑚 = 0 and 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} = σ_{𝑎𝑎𝑎𝑎}, we find 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_𝑎𝑎=𝜎𝜎_{𝑎𝑎𝑎𝑎}𝜀𝜀_{𝑎𝑎𝑎𝑎}

• Then, using Eqs. 14.5 and 14.4, we can define

𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_𝑎𝑎 = ℎ^′′ 𝑁𝑁_𝑓𝑓 (14.28)

𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 σ_𝑓𝑓^′

𝐸𝐸 2𝑁𝑁^𝑓𝑓

𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.29)

𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^{′ 2}

𝐸𝐸 2𝑁𝑁_𝑓𝑓 ^2𝑏𝑏 + σ_𝑓𝑓^′𝜀𝜀_𝑓𝑓^′ 2𝑁𝑁_𝑓𝑓 ^{𝑏𝑏+𝑐𝑐} (14.30)

14.3.5 Smith, Watson, and Topper (SWT) Parameters

Figure 14.13 Plot of the Smith, Watson, and Topper parameter versus life for the data of Fig. 14.10.

14.3.5 Walker Mean Stress Equation

• Recall Eq. (9.19)

where 𝑅𝑅 = 𝜎𝜎_{𝑚𝑚𝑚𝑚𝑛𝑛}/𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}

𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}^1−𝛾𝛾𝜎𝜎_𝑎𝑎^𝛾𝛾, 𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} 1 − 𝑅𝑅 (14.31)

2

𝛾𝛾

𝑁𝑁_𝑓𝑓 = 𝑁𝑁_{𝑚𝑚𝑚𝑚}^∗ 1 − 𝜎𝜎_𝑚𝑚 𝜎𝜎_𝑓𝑓^′

−1/𝑏𝑏

(14.23)

𝑁𝑁_𝑤𝑤^∗ = 𝑁𝑁_𝑓𝑓 𝜎𝜎_𝑎𝑎 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}

(1−𝛾𝛾)/𝑏𝑏

(14.32) 𝑁𝑁_𝑤𝑤^∗ = 𝑁𝑁_𝑓𝑓 1 − 𝑅𝑅

2

(1−𝛾𝛾)/𝑏𝑏

𝜀𝜀_𝑎𝑎 = σ_𝑓𝑓^′ 𝐸𝐸

1 − 𝑅𝑅 2

(1−𝛾𝛾)/𝑏𝑏

2𝑁𝑁_𝑓𝑓 ^𝑏𝑏 + 𝜀𝜀_𝑓𝑓^′ 1 − 𝑅𝑅 2

𝑐𝑐(1−𝛾𝛾)/𝑏𝑏

2𝑁𝑁_𝑓𝑓 ^𝑐𝑐 (14.33)

Backup

4.3 Strain-life Approach*

4.3.1 Notch Root Stresses and Strains

• Theoretical stress concentration factor, 𝐾𝐾_𝑡𝑡, is used to relate the nominal stress or strain to the local values.

• Upon yielding, the local values are no longer linearly related to the nominal values by 𝐾𝐾_𝑡𝑡

• Instead, the values are related in terms of stress and strain concentration factors as:

𝐾𝐾_𝜎𝜎 = Local stress

Nominal stress 𝐾𝐾_𝜖𝜖 = Local strain Nominal strain

(4.11)* (4.12)*

Figure 14.13* Effect of yielding on 𝐾𝐾_𝜎𝜎 and 𝐾𝐾_𝜖𝜖

* Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.

• After yielding, the actual local stress is less than that predicted using 𝐾𝐾_𝑡𝑡

• After yielding, the actual local strain is greater than that predicted using 𝐾𝐾_𝑡𝑡

• This is due to residual stresses at the notch root

14.1 Constant Amplitude Loading

• Let strain be expressed as a function of 𝑆𝑆 that denotes load, moment, nominal stress, etc.

𝜀𝜀 = 𝑙𝑙 𝑆𝑆 (14.47)

𝜀𝜀_{𝑚𝑚𝑎𝑎𝑚𝑚} = 𝑙𝑙 𝑆𝑆_{𝑚𝑚𝑎𝑎𝑚𝑚} = 𝑓𝑓 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} for tensile stress (14.48)

𝜀𝜀_𝑎𝑎 = 𝑙𝑙 𝑆𝑆_𝑎𝑎 = 𝑓𝑓 𝜎𝜎_𝑎𝑎 ∆𝜀𝜀

2 = 𝑙𝑙

∆𝑆𝑆

2 = 𝑓𝑓

∆𝜎𝜎 2